March 31, 2015

The 2015 ICC Cricket World Cup held in Australia and New Zealand saw over one million people attend its 49 matches, with attendances shown below, taken from here.

Match #

Date

Match

Venue

Attendance

1

14/02/15

New Zealand d Sri Lanka

Hagley Oval, Christchurch

17,228

2

14/02/15

Australia d England

Melbourne Cricket Ground

84,336

3

15/02/15

South Africa d Zimbabwe

Seddon Park, Hamilton

8,332

4

15/02/15

India d Pakistan

Adelaide Oval

41,587

5

16/02/15

Ireland d West Indies

Saxton Oval, Nelson

4,143

6

17/02/15

New Zealand d Scotland

University Oval, Dunedin

4,684

7

18/02/15

Bangladesh d Afghanistan

Manuka Oval, Canberra

10,972

8

19/02/15

Zimbabwe d UAE

Saxton Oval, Nelson

2,643

9

20/02/15

New Zealand d England

Westpac Stadium, Wellington

30,148

10

21/02/15

West Indies d Pakistan

Hagley Oval, Christchurch

14,461

11

21/02/15

Australia v Bangladesh

Gabba, Brisbane

washed out

12

22/02/15

Sri Lanka d Afghanistan

University Oval, Dunedin

2,711

13

22/02/15

India d South Africa

Melbourne Cricket Ground

86,876

14

23/02/15

England d Scotland

Hagley Oval, Christchurch

12,388

15

24/02/15

West Indies d Zimbabwe

Manuka Oval, Canberra

5,544

16

25/02/15

Ireland d UAE

Gabba, Brisbane

5,249

17

26/02/15

Afghanistan d Scotland

University Oval, Dunedin

3,229

18

26/02/15

Sri Lanka d Bangladesh

Melbourne Cricket Ground

30,012

19

27/02/15

South Africa d West Indies

Sydney Cricket Ground

23,612

20

28/02/15

New Zealand d Australia

Eden Park, Auckland

40,053

21

28/02/15

India d UAE

WACA Ground, Perth

8,718

22

1/3/2015

Sri Lanka d England

Westpac Stadium, Wellington

18,183

23

1/3/2015

Pakistan d Zimbabwe

Gabba, Brisbane

9,847

24

3/3/2015

South Africa d Ireland

Manuka Oval, Canberra

8,831

25

4/3/2015

Pakistan d UAE

McLean Park, Napier

2,406

26

4/3/2015

Australia d Afghanistan

WACA Ground, Perth

12,710

27

5/3/2015

Bangladesh d Scotland

Saxton Oval, Nelson

3,491

28

6/3/2015

India d West Indies

WACA Ground, Perth

17,557

29

7/3/2015

Pakistan d South Africa

Eden Park, Auckland

22,713

30

7/3/2015

Ireland d Zimbabwe

Blundstone Arena, Hobart

4,048

31

8/3/2015

New Zealand d Afghanistan

McLean Park, Napier

10,022

32

8/3/2015

Australia d Sri Lanka

Sydney Cricket Ground

39,951

33

9/3/2015

Bangladesh d England

Adelaide Oval

11,963

34

10/3/2015

India d Ireland

Seddon Park, Hamilton

10,192

35

11/3/2015

Sri Lanka d Scotland

Blundstone Arena, Hobart

3,549

36

12/3/2015

South Africa d UAE

Westpac Stadium, Wellington

4,901

37

13/03/15

New Zealand d Bangladesh

Seddon Park, Hamilton

10,347

38

13/03/15

England d Afghanistan

Sydney Cricket Ground

9,203

39

14/03/15

India d Zimbabwe

Eden Park, Auckland

30,076

40

14/03/15

Australia d Scotland

Blundstone Arena, Hobart

12,177

41

15/03/15

West Indies d UAE

McLean Park, Napier

1,221

42

15/03/15

Pakistan d Ireland

Adelaide Oval

9,889

43

18/03/15

QF1: South Africa d Sri Lanka

Sydney Cricket Ground

27,259

44

19/03/15

QF2: India d Bangladesh

Melbourne Cricket Ground

51,552

45

20/03/15

QF3: Australia d Pakistan

Adelaide Oval

35,516

46

21/03/15

QF4: New Zealand d West Indies

Westpac Stadium, Wellington

30,250

47

24/03/15

SF1: New Zealand d South Africa

Eden Park, Auckland

41,279

48

26/03/15

SF2: Australia d India

Sydney Cricket Ground

42,330

49

29/03/15

Final: Australia d New Zealand

Melbourne Cricket Ground

93,013

If we compare these numbers with the ground capacities shown below (largely taken from ground pages at ESPNcricinfo), we can make a graph of ground occupancy for each game.

Ground

Capacity

Melbourne Cricket Ground

95,000

Adelaide Oval

50,000

Sydney Cricket Ground

44,000

Eden Park, Auckland

41,000

Gabba, Brisbane

37,000

Westpac Stadium, Wellington

33,500

WACA Ground, Perth

24,500

Hagley Oval, Christchurch

18,000

Blundstone Arena, Hobart

16,200

Manuka Oval, Canberra

12,000

Seddon Park, Hamilton

12,000

McLean Park, Napier

10,500

Saxton Oval, Nelson

6,000

University Oval, Dunedin

5,000

We see that most of the first 10 matches (up to the wash-out between Australia and Bangladesh) were close to capacity, while some of the later matches leading up the quarter final had attendances well under 50% of capacity.

Finally, the table below shows the total attendances for games involving each country, as well as average ground occupancy percentages for their matches.

Team

total attendance

average % occupancy

#matches

attendance per match

New Zealand

277,024

94.2

9

30,780

Australia

360,086

83.7

8

45,011

India

288,888

73.8

8

36,111

South Africa

223,803

65.1

8

27,975

Scotland

39,518

63.7

6

6,586

Afghanistan

48,847

63.1

6

8,141

West Indies

96,788

60.4

7

13,827

Sri Lanka

138,893

58.6

7

19,842

England

166,221

57.8

6

27,704

Bangladesh

118,337

57.6

6

19,723

Pakistan

136,419

51.3

7

19,488

Ireland

42,352

47.8

6

7,059

Zimbabwe

60,490

47.4

6

10,082

UAE

25,138

23.8

6

4,190

Australia had a lowish attendance for its game against Afghanistan while New Zealand had every match at least 86% full.

Like this:

March 28, 2015

In this recent post, the following figure was formed by considering two triangular regions in three-dimensional space, where :

The intersection of the two triangles is the cross-section of a cube, but in this post we wish to explore further the centre of similarity of the two triangles.

The line joining and satisfies

Similarly, the line joining and satisfies

Equating the two expressions gives and from which . The point of intersection is therefore at . By symmetry of this expression the line joining and also passes through this point. This is the point on the plane that is equi-distant from the xy-, yz- and xz- coordinate planes. It is also the central projection of the origin onto the plane along the vector parallel to .

In terms of the original two triangles this point is neither the centroid, incentre, orthocentre, circumcentre nor other commonly encountered triangle centre. Let us find the barycentric coordinates of this point (call it ) in terms of the triangle with vertices at .

The first barycentric coordinate will be the ratio of the area of to the area of . Since and have the same x-coordinate, this will be the ratio of the x-coordinates of to , which is . By symmetry it follows that the barycentric coordinates have the attractive form

Let the side lengths of be . Then by Pythagoras’ theorem, . Hence

.

By the cosine rule, (where ) which equals from the above expression. Therefore and similarly we obtain . Then

By the sine rule, ( being the circumradius of ) from which . Hence the barycentric coordinates of may be written in non-normalised form as

Comparing this with the coordinates of the orthocentre , the point is known as the square root of the orthocentre (see Theorem 1 of [1]). Note that the real existence of the point requires to be acute, which it is when . A geometric construction of the square root of a point is given in Section 8.1.2 of [2].